Relationship between exponentials & logarithms: tables

Voiceover:We've got 2 tables over here. This first table tells us that for any given value of X, what is the value of b to the x? For example, if we look right over here, if x is 1.585, b to the 1.585 is 3. This is telling us that b to the 1.585 is equal to 3. Similarly ... I can never say that ... it is telling us that b to the 2.322 is 5. b to the 2.807 is 7. b to the 2.169 is 9. Now this table over here is telling us for any given value of y, what is going to be log base b of y? This tells us that log base b of a is 0. Log base b of 2 is 1. Log base b of 2c is 1.585. Log base b of 10d, so this is literally, this is telling us that log base b of 10d is equal to 2.322. That's what this last column tells us. Now what I challenge you to do is pause this video, and using just the information here, and you don't need a calculator; in fact, you can't use a calculator. I forbid you; try to figure out what a, b, c, and d are just using your powers of reasoning, no calculator involved. Just use your powers of reasoning. Can you figure out what a, b, c, and d are? I'm assuming you've given a go at it, so let's see what we can deduce from this. Here, we have just a bunch of numbers. We need to figure out what b is. These are all b to the 1.585 power is 3. I don't really know what to make sense of this stuff here. Maybe this table will help us. This first ... Let me do these in different colors. This first column right over here tells us that log base b of a, so now y is equal to a, that that is equal to 0. Now this is an equivalent statement to saying that b to the a power is equal to ... oh sorry, not b to the a power. This is an equivalent statement to saying b to the 0 power is equal to a. This is saying what exponent do I need to raise b to to get a? You raise it to the 0 power. This is saying b to the 0 power is equal to a. Now what is anything to the 0 power, assuming that it's not 0? If we're assuming that b is not 0, if we're assuming that b is not 0, so we're going to assume that, and we can assume, and I think that's a safe assumption because where we're raising b to all of these other powers, we're getting a non-0 value. Since we know that b is not 0, anything with a 0 power is going to be 1. This tells us that a is equal to 1. We got one figured out. a is equal to 1. Now let's look at this next piece of information right over here. What does that tell us? That tells us that log base b of 2 is equal to 1. This is equivalent to saying the power that I needed to raise b to get to to 2 is 1. Or if I want to write in exponential form, I could write this as saying that b to the first power is equal to 2. I'm raising something to the first power and I'm getting 2? What is this thing? That means that b must be 2. 2 to the first power is 2. So b is equal to 2. b to the first power is equal to 2. You could say b to the first is equal to 2 to the first. That's also equal to 2. So b must be equal to 2. We've been able to figure that out. This is a 2 right over here. It actually makes sense. 2 to the 1.585 power, yeah, that feels right, that that's about 3. Now let's see what else we can do. Let's see if we can figure out c. Let's look at this column. Let's see what this column is telling us. That column we could read as log base b. Now our y is 2c. Log base b of 2c is equal to 1.585. Or we could read this as b, if we write in exponential form, b to the 1.585 is equal to 2c. Now what's b to the 1.585? They tell us right over here that b to the 1.585 is 3, is 3, so this right over here is equal to 3. We get 2c is equal to 3, or divide both sides by 2, we would get c is equal to 1.5. This is working out pretty well. Now we have this last column, which I will circle in purple, and we can write this as log base b of 10d is equal to 2.322. This is saying the power I need to raise b to to get to 10d is 2.322, or in exponential form, b to the 2.322 is equal to 10d. Now what is b to the 2.322? They tell us over here. b to the 2.322 is 5. This is equal to 5. We could write 10d is equal to 5, or divide both sides by 10. d is equal to 0.5, and we're done. We were able to figure out what a, b, c, and d are without the use of a calculator.